The smti (smti) is a special case of spa-st in which each project and lecturer has capacity 1, and each lecturer offers one project. Given an instance of smti, it is known that stable matchings can have different sizes [1]}, and thus the same is true for spa-st. Yet in practical applications it is desirable to match as many students to projects as possible. This motivates max-spa-st, the problem of finding a maximum (cardinality) stable matching in an instance of spa-st. This problem is NP-hard, since the corresponding optimisation problem restricted to smti, which we refer to as max-smti, is NP-hard [1]}. Király [3]} described a \(\frac{3}{2}\) -approximation algorithm for max-smti. He also showed how to extend this algorithm to the case of the hrt (hrt), where hrt is the special case of spa-st in which each lecturer \(l\)  offers one project \(p\) , and the capacities of \(l\)  and \(p\)  are equal. Yanagisawa [4]} showed that max-smti is not approximable within a factor of \(\frac{33}{29}\)  unless P=NP; the same bound applies to max-spa-st.
